Swarnava Mukhopadhyay




Presently I am thinking about questions related to semiorthogonal decomposition of bounded derived categories, modular functors and braided tensor categories, Verlinde type formulas for twisted conformal blocks, moduli of parahoric Bruhat-Tits torsors and their connections to conformal blocks. I am also interested in questions related to hyperplane arrangements and in particular its connection to representation theory of Lie algebras. I have also worked on applying conformal blocks divisor to the birational geometry of moduli of curves.

A brief presentation of my recent work can be found here. Please feel free to browse my papers below. My past and current collaborators are P. Belkale, P. Belmans, I. Biswas, P. Brosnan,T. Deshpande, S. Galkin, A. Gibney, A. Paul, R. Wentworth. and H. Zelaci

Here is the link to the algebraic geometry seminar in TIFR. Along with Chiara Damiolini and Johan Martens, I organized a conference on Bundles and Conformal Blocks with a twist in ICMS, Edinburgh from June 13-17, 2022

Publications/Preprints

  1. Combinatorial non-abelian Torelli theorem
    with P. Belmans and S. Galkin. Click here for the present version.
    [Abstract]

    We prove that a (colored) trivalent graph can be recovered from (the polar dual of) the associated quantum Clebsch-Gordan polytope and that any isomorphism between such polytopes is induced by a unique properly defined isomorphism of underlying colored graphs. We also show how graph potentials introduced by the authors relate to the theory of random walks, and we use our combinatorial Torelli theorem to construct random walks with distinct shapes but equal return probabilities for every number of steps


  2. Graph potentials and symplectic geometry of the moduli space of vector bundles
    with P. Belmans and S. Galkin, arXiv:2206.11584, separated from arXiv:2009.05568 along with proof of conjecture on equality of periods. Click here for slides of a talk. Click here for video of a talk.
    [Abstract] For any monotone Lagrangian torus L on a Fano manifold X the weighted number of holomorphic Maslov index 2 discs with boundary on L is bounded from below by an invariant TX (introduced earlier in the Galkin–Golyshev–Iritani conjecture). If this bound is attained the torus L is said to be optimal. We give the first examples of Fano manifolds with multiple optimal tori. To every trivalent graph $\gamma$ of genus g we associate an optimal torus L on the celebrated symplectic Fano manifold Ng (of complex dimension 3g - 3 with TNg = 8g- 8), namely the character variety of rank 2 on a genus g surface with prescribed odd monodromy at a puncture, and we show that all pairs N_g; L_{\gamma} are pairwise non-isotopic. In particular, we confirm a form of mirror symmetry between the A-model of these pairs N_g; L_{\gamma} and B-model of a subclass of graph potentials, a family of Laurent polynomials we introduced in earlier work. A crucial input from outside of symplectic geometry is an analysis of Manon’s toric degenerations of algebro-geometric models for the spaces Ng, as moduli spaces of stable rank 2 bundles on an algebraic curve with a fixed determinant, constructed using conformal field theory.



  3. Graph potentials and topological quantum field theories
    with P. Belmans and S. Galkin, arXiv:2205.07244. Click here for slides of a talk. Click here for video of a talk.
    [Abstract]

    We introduce graph potentials, which are Laurent polynomials associated to (colored) trivalent graphs. We show that the birational type of the graph potential only depends on the homotopy type of the colored graph, and thus define a topological quantum field theory. We end by giving an efficient computational method to compute its partition function.


  4. Decompositions of the moduli space of vector bundles and graph potentials
    with P. Belmans and S. Galkin,formerly part of arXiv:2009.05568 along with new material on dimension of critical loci. Click here for slides of a talk. Click here for video of a talk.
    [Abstract]

    We propose a conjectural semiorthogonal decomposition for the derived category of the moduli space of stable rank 2 bundles with fixed determinant of odd degree, independently formulated by Narasimhan. We discuss some evidence for, and furthermore propose semiorthogonal decompositions with additional structure. We also discuss two other decompositions.One is a decomposition of this moduli space in the Grothendieck ring of varieties, which relates to various known motivic decompositions. The other is the critical value decomposition of a candidate mirror Landau– Ginzburg model given by graph potentials, which in turn is related under mirror symmetry to Muñoz’s decomposition of quantum cohomology. This corresponds to an orthogonal decomposition of the Fukaya category. We will explain how these decompositions can be seen as evidence for the conjectural semiorthogonal decomposition.


  5. Geometrization of the Hitchin/WZW/KZ connection
    with I. Biswas and R. Wentworth, arXiv:2110.00430 , 38 Pages. Click here for video of a talk by Richard Wentworth.
    [Abstract]

    Given a simple, simply connected, complex algebraic group G, a flat projective connection on the bundle of nonabelian theta functions on the moduli space of semistable parabolic G-bundles over any family of smooth projective curves with marked points was constructed by the authors in an earlier paper. Here, it is shown that the identification between the bundle of nonabelian theta functions and the bundle of WZNW conformal blocks is flat with respect to this connection and the one constructed by Tsuchiya-Ueno-Yamada. As an application, we give a geometric construction of the Knizhnik-Zamolodchikov connection on the trivial bundle over the configuration space of points in the projective line whose typical fiber is the space of invariants of tensor product of representations.


  6. Ginzburg algebras and the Hitchin connection for parabolic G-bundles
    with I. Biswas and R. Wentworth, arXiv:2103.03792, 57 Pages. Click here for video of a talk.
    [Abstract]

    For a simple, simply connected, complex group G, we prove the existence of a flat projective connection on the bundle of nonabelian theta functions on the moduli space of semistable parabolic G-bundles over families of smooth projective curves with marked points.


  7. Appendix D: Rank-level duality of conformal blocks-A brief survey
    Book by Shrawan Kumar entitled: Conformal blocks, generalized theta functions and the Verlinde formula, New Mathematical Monograph series, Cambridge University Press, 2021, 15 Pages.
    [Abstract]

    This is a brief survey on rank-level duality results of conformal blocks on curves. This will appear as an appendix to Shawan Kumar's book.


  8. Crossed modular categories and the Verlinde formula for twisted conformal blocks
    with T. Deshpande, arXiv:1909.10799, 107 Pages. Click here for slides of a talk.
    [Abstract]

  9. Spectral data for spin Higgs bundles
    with R. Wentworth, Preprint arXiv:1809.05516
    International Math. Research Notices no.6 2021, 20 Pages.
    [Abstract]

  10. Conformal Embeddings and twisted theta functions at level one
    with H. Zelaci
    in Proceedings of the American Mathematical Society 148 (2020), 14 Pages.
    [Abstract]

  11. Examples violating Golyshev's canonical strip hypothesis
    with P. Belmans and S. Galkin
    to appear in Experimental Mathematics 31 (2022), no. 1, 233–237, Preprint arXiv:1806.07648, 8 Pages.
    [Abstract]

  12. Admissble subcategories in the derived category of moduli spaces of vector bundles on curves.
    with P. Belmans
    Advances in Mathematics, 351(2019), Preprint arXiv:180700216. 22 Pages.
    [Abstract]

  13. Topology of hyperplane arrangements and invariant theory.
    with P. Belkale and P. Brosnan.
    Michigan Mathematical Journal 68(2019), Preprint arXiv:1611.01861. 37 Pages.
    [Abstract]

  14. On higher Chern classes of vector bundles of conformal blocks
    with A. Gibney. Preprint 13 Pages.
    [Abstract]

  15. Fundamental groups of moduli spaces of Principal bundles over a curve .
    with I. Biswas and A. Paul.
    in Geometriae Dedicata 214(2021), Preprint arXiv:1609.06436, 12 Pages.
    [Abstract]

  16. Generalized theta functions, strange duality, and odd orthogonal bundles on curves.
    with R. Wentworth.
    Communications in Mathematical Physics, 370 (2019) no.1, Preprint arXiv:1608.04990, 51 pages.
    [Abstract]

  17. Strange duality of Verlinde spaces for G_2 and F_4.
    Mathematische Zeitschrift 283 (2016), no 1-2, arXiv:1504.03757, 13 Pages.
    [Abstract]

  18. Rank-level duality and conformal blocks divisors.
    Published in Adv. Math. Volume 287, 10 January 2016, arXiv:1308.0854, 23 Pages.
    [Abstract]

  19. Non-vanishing of conformal blocks divisors on \bar-M_{0,n}.
    with P. Belkale and A. Gibney
    Transformation Groups 21(2016), no-2, arXiv:1410.2459, 25 Pages.
    [Abstract]

  20. Vanishing and identities of conformal blocks divisors.
    with P. Belkale and A. Gibney
    Published in Algebr. Geom. 2 (2015), no. 1 arXiv:1308.4906, 29 Pages.
    [Abstract]

  21. Rank-level duality of conformal blocks for odd orthogonal Lie algebras in genus 0.
    Trans. Amer. Math. Soc. 368 (2016), no. 9, arXiv:1211.2204, 38 Pages.
    [Abstract]

  22. Conformal blocks to cohomology in genus zero.
    with P. Belkale
    Published in Ann. Inst. Fourier 64(2014). arXiv:1205.3199, 52 Pages.
    [Abstract]

  23. Remarks on level one conformal blocks divisors.
    Published in C. R. Math. Acad. Sci. Paris 352 (2014), no. 3, 4 Pages.
    [Abstract]

  24. Diagram automorphisms and rank-level duality.
    arXiv:1308.1756, 17 Pages.(part of this has appeared in my thesis Trans. Amer. Math. Soc. 368 (2016), no. 9 and also in Appedix D of the book of Shrawan Kumar).
    [Abstract]

Thesis and Unpublished Notes

  1. Phd Thesis: Rank-level duality of conformal blocks.
    under the direction of P. Belkale.
    Click here to view. [abstract]

  2. Masters Paper: Factorization of conformal blocks.
    under the direction of S. Kumar.
    Click here to view. [abstract]

  3. Rank-level duality of conformal blocks for SL_n in genus 0 revisited.
    Unpublished. We give an alternate proof of Nakanishi-Tsuchiya's rank-level duality theorem. Notes available on request. [abstract]

  4. Notes on action of diagram automorphism on conformal blocks divisors
    Not for publication. Click here for a copy. [abstract]